Subgroups — C₂×#106×Dic₁₃ #15860960

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
4	#106×Dic₁₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718121	C₂	1	2	yes	maximal
6	C₂²×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₁₃	—	2	2	no	—
11	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	C₄	1	8	yes	complemented
14	C₂×C₁₃×#106	C₂²	1	4	yes	—
15	C₂×#88071	—	2	2	no	—
16	C₂×#88057	C₄	1	8	yes	complemented
17	C₂×#88057	C₂²	1	4	yes	—
18	C₂×C₈×Dic₁₃	C₄	1	8	yes	complemented
19	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₁₃	—	2	4	no	—
21	C₄⋊C₄×Dic₁₃	C₂²	1	8	yes	—
22	#11718121	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	—	2	2	no	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
27	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	4	no	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
40	C₂×C₄×Dic₁₃	—	2	8	no	—
41	C₂×C₄×Dic₁₃	—	4	8	no	—
42	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
44	C₂×C₄×Dic₁₃	Q₈	1	4	yes	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×#1055	—	2	2	no	—
47	C₂²×#1055	—	2	2	no	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₂²×#1055	C₂³	1	2	yes	—
50	C₁₃×#106	C₂×C₄	1	16	yes	complemented
51	#88071	—	2	8	no	—
52	#88057	C₂×C₄	1	16	yes	complemented
53	C₈×Dic₁₃	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×#106	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	—	4	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	4	no	—
69	C₂²×Dic₁₃	—	2	8	no	—
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	—	4	4	no	—
72	C₂²×Dic₁₃	—	4	4	no	—
73	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
74	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
75	C₂²×Dic₁₃	SD₁₆	1	8	yes	—
76	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
77	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₁₃×C₄⋊C₄	—	2	8	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
85	C₄⋊Dic₁₃	—	2	8	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
88	C₄×Dic₁₃	—	4	16	no	—
89	C₄×Dic₁₃	C₄⋊C₄	1	8	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
91	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
92	C₂×#1055	—	2	8	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	C₂²×C₄	1	4	yes	—
96	C₂²×#106	—	13	1	no	—
97	C₂²×#106	D₁₃	1	1	yes	characteristic
98	C₂²×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₄⋊C₄	—	13	2	no	—
100	C₄×#106	—	13	8	no	—
101	C₂×#340	—	13	2	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₁₃	C₄×D₄	1	2	yes	—
104	C₂³×C₁₃	C₄×D₄	1	2	yes	—
105	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
106	C₂³×C₁₃	#135	1	4	yes	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	16	no	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	4	no	—
126	C₂×Dic₁₃	—	2	8	no	—
127	C₂×Dic₁₃	—	2	8	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	—	4	4	no	—
130	C₂×Dic₁₃	—	4	16	no	—
131	C₂×Dic₁₃	#106	1	16	yes	direct product factor
132	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×Dic₁₃	C₂×SD₁₆	1	8	yes	—
135	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
136	#1055	—	2	4	no	—
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	13	2	no	—
140	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
141	C₂²×C₄²	—	13	1	no	—
142	C₂²×C₄²	—	26	2	no	—
143	C₂×#106	—	13	8	no	—
144	C₂×#106	—	13	4	no	—
145	C₂×#106	Dic₁₃	1	8	yes	direct product factor
146	C₂×#106	C₂×D₁₃	1	4	yes	—
147	C₂×C₄×C₈	—	13	8	no	—
148	C₂×C₄×C₈	—	13	4	no	—
149	C₄×C₄⋊C₄	—	13	8	no	—
150	C₄×C₄⋊C₄	—	26	4	no	—
151	C₂×#95	—	26	2	no	—
152	#340	—	13	8	no	—
153	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
154	C₂²×C₁₃	C₂×#106	1	4	yes	—
155	C₂²×C₁₃	#427	1	4	yes	—
156	C₂²×C₁₃	C₄×SD₁₆	1	8	yes	—
157	C₂²×C₁₃	C₄×SD₁₆	1	8	yes	—
158	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
162	C₂²×C₁₃	C₂×#135	1	2	yes	—
163	C₄×C₁₃	—	4	8	no	—
164	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
165	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	Dic₁₃	—	2	4	no	—
170	Dic₁₃	—	4	8	no	—
171	Dic₁₃	C₂×#106	1	8	yes	direct product factor
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	13	1	no	—
174	C₂³×C₄	—	13	1	no	—
175	C₂³×C₄	—	26	2	no	—
176	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	—	13	4	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	—	26	2	no	—
181	C₂²×C₈	C₄×D₁₃	1	4	yes	—
182	C₂²×C₈	C₄×D₁₃	1	4	yes	—
183	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
184	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	13	8	no	—
188	C₂×C₄⋊C₄	—	13	4	no	—
189	C₂×C₄⋊C₄	—	26	2	no	—
190	C₂×C₄⋊C₄	—	26	4	no	—
191	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
192	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
193	C₂×C₄²	—	13	4	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	4	no	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	26	8	no	—
198	C₂×C₄²	—	52	8	no	—
199	#106	—	13	16	no	—
200	#106	C₂×Dic₁₃	1	16	yes	direct product factor
201	C₄×C₈	—	13	16	no	—
202	#95	—	26	8	no	—
203	C₂×C₁₃	C₄×#106	1	8	yes	complemented
204	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
207	C₂×C₁₃	C₂×C₄×SD₁₆	1	2	yes	—
208	C₂×C₁₃	C₂×C₄×SD₁₆	1	2	yes	—
209	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	13	4	no	—
215	C₂²×C₄	—	13	8	no	—
216	C₂²×C₄	—	13	2	no	—
217	C₂²×C₄	—	13	4	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	26	4	no	—
220	C₂²×C₄	—	26	8	no	—
221	C₂²×C₄	—	26	4	no	—
222	C₂²×C₄	—	52	4	no	—
223	C₂²×C₄	—	52	4	no	—
224	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
226	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	#6129	1	2	yes	—
228	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
229	C₂²×C₄	#6131	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
231	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	26	8	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
237	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
238	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
239	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	13	8	no	—
243	C₄⋊C₄	—	26	8	no	—
244	C₄⋊C₄	—	26	4	no	—
245	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
246	C₄²	—	13	8	no	—
247	C₄²	—	13	4	no	—
248	C₄²	—	13	4	no	—
249	C₄²	—	52	16	no	—
250	C₁₃	C₂×C₄×#106	1	1	yes	characteristic, complemented, Sylow
251	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
252	C₂³	#88147	1	2	yes	—
253	C₂³	SD₁₆×D₁₃	1	4	yes	—
254	C₂³	#88170	1	4	yes	—
255	C₂³	D₄×Dic₁₃	1	2	yes	—
256	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	13	4	no	—
262	C₂×C₄	—	13	8	no	—
263	C₂×C₄	—	13	16	no	—
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	26	4	no	—
266	C₂×C₄	—	26	8	no	—
267	C₂×C₄	—	26	8	no	—
268	C₂×C₄	—	52	4	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	52	16	no	—
271	C₂×C₄	#88141	1	4	yes	—
272	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
273	C₂×C₄	#88146	1	4	yes	—
274	C₂×C₄	#88147	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
276	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
277	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
278	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#6129	1	1	yes	characteristic
281	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#6131	1	1	yes	characteristic
283	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
284	C₈	—	26	4	no	—
285	C₈	C₂×C₄×Dic₁₃	1	8	yes	complemented
286	C₂²	#11718513	1	8	yes	—
287	C₂²	D₁₃×#106	1	4	yes	—
288	C₂²	#11718521	1	4	yes	—
289	C₂²	C₄⋊C₄×Dic₁₃	1	4	yes	—
290	C₂²	SD₁₆×Dic₁₃	1	8	yes	—
291	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
292	C₂²	C₂×#88147	1	1	yes	characteristic
293	C₂²	C₂×SD₁₆×D₁₃	1	2	yes	—
294	C₂²	C₂×#88170	1	2	yes	—
295	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
296	C₄	—	4	8	no	—
297	C₄	—	13	8	no	—
298	C₄	—	26	4	no	—
299	C₄	—	52	8	no	—
300	C₄	C₄⋊C₄×Dic₁₃	1	4	yes	—
301	C₄	C₂×#88141	1	1	yes	characteristic
302	C₄	C₂×#88146	1	1	yes	characteristic
303	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
304	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
305	C₂	#106×Dic₁₃	1	8	yes	direct product factor
306	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
307	C₂	C₂×D₁₃×#106	1	1	yes	characteristic
308	C₂	C₂×#11718521	1	1	yes	characteristic
309	C₂	C₂×#11718513	1	2	yes	—
310	C₂	C₂×SD₁₆×Dic₁₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#106	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#106	D₁₃
hypocenter	C₁₃	C₂×C₄×#106
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #106×Dic₁₃	C₂	direct (N × H)
13. C₂×C₁₃×#106	C₄	semidirect
16. C₂×#88057	C₄	semidirect
18. C₂×C₈×Dic₁₃	C₄	semidirect
50. C₁₃×#106	C₂×C₄	semidirect
52. #88057	C₂×C₄	semidirect
53. C₈×Dic₁₃	C₂×C₄	semidirect
78. C₂×C₈×C₁₃	C₄²	semidirect
131. C₂×Dic₁₃	#106	direct (N × H)
135. C₈×C₁₃	C₂×C₄²	semidirect
145. C₂×#106	Dic₁₃	direct (N × H)
171. Dic₁₃	C₂×#106	direct (N × H)
200. #106	C₂×Dic₁₃	direct (N × H)
203. C₂×C₁₃	C₄×#106	semidirect
236. C₂×C₈	C₄×Dic₁₃	semidirect
250. C₁₃	C₂×C₄×#106	semidirect
285. C₈	C₂×C₄×Dic₁₃	semidirect
305. C₂	#106×Dic₁₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#88057	2
C₂²×C₈×Dic₁₃	2
C₂×C₄⋊C₄×Dic₁₃	2
#106×Dic₁₃	2
C₂×#11718121	2
C₂²×C₁₃×#106	2
C₂×C₄×#106	13

Subgroups — C2×#106×Dic13 #15860960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#106×Dic₁₃ #15860960